Properties

Degree 2
Conductor $ 2^{4} \cdot 3 \cdot 5^{2} \cdot 13 \cdot 17 $
Sign $-1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 1

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s − 3·7-s + 9-s + 2·11-s + 13-s + 17-s − 2·19-s + 3·21-s − 4·23-s − 27-s + 5·29-s − 8·31-s − 2·33-s + 3·37-s − 39-s + 8·41-s − 13·43-s − 3·47-s + 2·49-s − 51-s + 2·57-s − 4·59-s − 14·61-s − 3·63-s + 2·67-s + 4·69-s − 14·71-s + ⋯
L(s)  = 1  − 0.577·3-s − 1.13·7-s + 1/3·9-s + 0.603·11-s + 0.277·13-s + 0.242·17-s − 0.458·19-s + 0.654·21-s − 0.834·23-s − 0.192·27-s + 0.928·29-s − 1.43·31-s − 0.348·33-s + 0.493·37-s − 0.160·39-s + 1.24·41-s − 1.98·43-s − 0.437·47-s + 2/7·49-s − 0.140·51-s + 0.264·57-s − 0.520·59-s − 1.79·61-s − 0.377·63-s + 0.244·67-s + 0.481·69-s − 1.66·71-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 265200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 265200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(265200\)    =    \(2^{4} \cdot 3 \cdot 5^{2} \cdot 13 \cdot 17\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{265200} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(2,\ 265200,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3,\;5,\;13,\;17\}$,\[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;3,\;5,\;13,\;17\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + T \)
5 \( 1 \)
13 \( 1 - T \)
17 \( 1 - T \)
good7 \( 1 + 3 T + p T^{2} \)
11 \( 1 - 2 T + p T^{2} \)
19 \( 1 + 2 T + p T^{2} \)
23 \( 1 + 4 T + p T^{2} \)
29 \( 1 - 5 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 - 3 T + p T^{2} \)
41 \( 1 - 8 T + p T^{2} \)
43 \( 1 + 13 T + p T^{2} \)
47 \( 1 + 3 T + p T^{2} \)
53 \( 1 + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 + 14 T + p T^{2} \)
67 \( 1 - 2 T + p T^{2} \)
71 \( 1 + 14 T + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 - 2 T + p T^{2} \)
83 \( 1 - 11 T + p T^{2} \)
89 \( 1 + 2 T + p T^{2} \)
97 \( 1 - 17 T + p T^{2} \)
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\[\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−13.04595972253089, −12.52811293288334, −12.01536258643091, −11.85566976465099, −11.14260614473851, −10.69942186414661, −10.33027017229311, −9.718016047579848, −9.489403911148899, −8.894767999344925, −8.480795942387596, −7.691199411845688, −7.476931392335138, −6.594125770581111, −6.452459788901673, −6.067584917915149, −5.486385003325715, −4.901687308714493, −4.257841841041489, −3.895448308363080, −3.219013706404672, −2.871808557156751, −1.909728657824197, −1.498168488432346, −0.6142685557471942, 0, 0.6142685557471942, 1.498168488432346, 1.909728657824197, 2.871808557156751, 3.219013706404672, 3.895448308363080, 4.257841841041489, 4.901687308714493, 5.486385003325715, 6.067584917915149, 6.452459788901673, 6.594125770581111, 7.476931392335138, 7.691199411845688, 8.480795942387596, 8.894767999344925, 9.489403911148899, 9.718016047579848, 10.33027017229311, 10.69942186414661, 11.14260614473851, 11.85566976465099, 12.01536258643091, 12.52811293288334, 13.04595972253089

Graph of the $Z$-function along the critical line